Math::...licate(3pm)

📦
version 129 · 3 versions
, in debian bookworm, debian trixie, ubuntu jammy, ubuntu noble, ubuntu resolute, ubuntu stonking

✨ NAME ✨

Math::PlanePath::GosperReplicate — self-similar hexagon replications 💖

(Yes bestie, this is one of those User Contributed Perl Documentation pages, and it is so much cooler than the name makes it sound.)

🌸 SYNOPSIS 🌸

use Math::PlanePath::GosperReplicate;
my $path = Math::PlanePath::GosperReplicate->new;
my ($x, $y) = $path->n_to_xy (123);

🦋 DESCRIPTION 🦋

Okay so picture this: a self-similar hexagonal tiling of the plane. Every single level, the shape is the Gosper island. Iconic behaviour, honestly. 🏝️

                            17----16                     4
                           /        \
             24----23    18    14----15                  3
            /        \     \
          25    21----22    19----20    10---- 9         2
            \                          /        \
             26----27     3---- 2    11     7---- 8      1
                        /        \     \
          31----30     4     0---- 1    12----13     <- Y=0
         /        \     \
       32    28----29     5---- 6    45----44           -1
         \                          /        \
          33----34    38----37    46    42----43        -2
                     /        \     \
                   39    35----36    47----48           -3
                     \
                      40----41                          -4

                             ^
       -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7

So the points get spread out on every second X coordinate so that it makes a triangular lattice in integer coordinates — that's the "Triangular Lattice" thing in Math::PlanePath, iykyk. 💅

The base pattern is the inner N=0 to N=6, and then six copies of that shape get arranged around it as the blocks N=7,14,21,28,35,42. Then six copies of the resulting N=0 to N=48 shape get replicated around, and on and on, forever, periodt. 🔁

Every point can be taken as a little hexagon, so that all the points tile the plane with hexagons. The innermost N=0 to N=6 look like this:

         *     *
        / \   / \
       /   \ /   \
      *     *     *
      |  3  |  2  |
      *     *     *
     / \   / \   / \
    /   \ /   \ /   \
   *     *     *     *
   |  4  |  0  |  1  |
   *     *     *     *
    \   / \   / \   /
     \ /   \ /   \ /
      *     *     *
      |  5  |  6  |
      *     *     *
       \   / \   /
        \ /   \ /
         *     *

The further replications are the same arrangement, except the sides become ever wigglier and the centres rotate around. You can literally see the rotation at N=7 at X=5,Y=1, which is up from the X axis. 🌀

The "FlowsnakeCentres" path is this same replicating shape, but it starts from a side instead of the middle and traverses in such a way as to make each N adjacent. And the "Flowsnake" curve itself is this replication too, but with segments across hexagons. 🌈

🔢 Complex Base

This path corresponds to expressing complex integers X+i*Y in a base

    b = 5/2 + i*sqrt(3)/2

with coordinates scaled to put equilateral triangles on a square grid. So for integer X,Y on the triangular grid (where X,Y are either both odd or both even),

    X/2 + i*Y*sqrt(3)/2 = a[n]*b^n + ... + a[2]*b^2 + a[1]*b + a[0]

where each digit a[i] is either 0 or a sixth root of unity encoded into base-7 digits of N,

     w6 = e^(i*pi/3)            sixth root of unity, b = 2 + w6
        = 1/2 + i*sqrt(3)/2

     N digit     a[i] complex number
     -------     -------------------
       0          0
       1         w6^0 =  1
       2         w6^1 =  1/2 + i*sqrt(3)/2
       3         w6^2 = -1/2 + i*sqrt(3)/2
       4         w6^3 = -1
       5         w6^4 = -1/2 - i*sqrt(3)/2
       6         w6^5 =  1/2 - i*sqrt(3)/2

7 digits suffice because

     norm(b) = (5/2)^2 + (sqrt(3)/2)^2 = 7

🔄 Rotate Numbering

The parameter "numbering_type => 'rotate'" applies a rotation in each sub-part according to its location around the preceding level. ✨

You can see the effect by writing N in base-7. Part 10-16 is the same as the middle 0-6. Part 20-26 has a rotation by +60 degrees. Part 30-36 has rotation by +120 degrees, and so on. 💫

                                22----21
                               /     /           numbering_type => 'rotate'
                 31    36    23    20    26          N shown in base-7
                /  \     \     \        /
              32    30    35    24----25    13----12
                \        /                 /        \
                 33----34     3---- 2    14    10----11
                            /        \     \
              46----45     4     0---- 1    15----16
                      \     \
           41----40    44     5---- 6    64----63
             \        /                 /        \
              42----43    55----54    65    60    62
                         /        \     \     \  /
                       56    50    53    66    61
                            /     /
                          51----52

Notice how this means in each part the 11, 21, 31, etc, points are directed away from the middle in the same way, relative to the sub-part locations. 👀

Working through the expansions gives the following rule for when an N is on the boundary of level k,

    write N in k many base-7 digits  (empty string if k=0)
    if any 0 digit then non-boundary
    ignore high digit and all 1 digits
    if any 4 or 5 digit then non-boundary
    if any 32, 33, 66 pair then non-boundary

A 0 digit is the middle of a block, or a 4 or 5 digit the inner side of a block, for k>=1, hence non-boundary. After that the 6,1,2,3 parts variously expand with rotations so that a 66 is enclosed on the clockwise side and 32 and 33 on the anti-clockwise side. 🔍

🛠️ FUNCTIONS 🛠️

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes. 🫶

  • $path = Math::PlanePath::GosperReplicate->new ()
  • $path = Math::PlanePath::GosperReplicate->new (numbering_type => $str) — Create and return a new path object. The "numbering_type" parameter can be
    "fixed"        (default)
    "rotate"
  • ($x,$y) = $path->n_to_xy ($n) — Return the X,Y coordinates of point number $n on the path. Points begin at 0 and if "$n < 0" then the return is an empty list. 🕳️

📏 Level Methods

  • ($n_lo, $n_hi) = $path->level_to_n_range($level) — Return "(0, 7**$level - 1)".

📐 FORMULAS 📐

🔄 Axis Rotations

In the fixed numbering, digit positions 1,2,3,4,5,6 go around +60deg each, so the N for rotation of X,Y by +60 degrees is each digit +1.

    N          = 0, 1, 2, 3, 4, 5, 6, 10, 11, 12

    rot+60(N)  = 0, 2, 3, 4, 5, 6, 1, 14, 16, 17, ... decimal
               = 0, 2, 3, 4, 5, 6, 1, 20, 22, 23, ... base7

    rot+120(N) = 0, 3, 4, 5, 6, 1, 2, 21, 24, 25, ... decimal
               = 0, 3, 4, 5, 6, 1, 2, 30, 33, 34, ... base7

    etc

In the rotate numbering, just adding +1 (etc) at the high digit alone is rotation. 💅

📏 X,Y Extents

The maximum X in a given level N=0 to 7^k-1 can be calculated from the replications. A given high digit 1 to 6 has sub-parts located at b^k*w6^(d-1). Those sub-parts are all the same, so the one with maximum real(b^k*w6^(d-1)) contains the maximum X. 👑

    N_xmax_digit(j) = d=1to6 where real(w6^(d-1) * b^j) is maximum
                    = 1,1,6,6,6,5,5,5,4,4,4,3,3,3,3,2,2, ...

                 k-1
    N_xmax(k) = digits N_xmax_digit(j)    low digit j=0
                 j=0
              = 0, 1, 8, 302, 2360, 16766, 100801, ...  decimal
              = 0, 1, 11, 611, 6611, 66611, 566611, ...  base7

                k-1
    z_xmax(k) = sum  w6^d[j] * b^j
                j=0      each d[j] with real(w6^d[j] * b^j) maximum
              = 0, 1, 7/2+1/2*sqrt3*i, 10-sqrt3*i, 57/2-3/2*sqrt3*i,...

    xmax(k) = 2*real(z_xmax(k))
            = 0, 2, 7, 20, 57, 151, 387, 1070, 2833, 7106, ...

For computer calculation these maximums can be calculated from the powers. The parts resulting can also be written in terms of the angle

    arg(b) = atan(sqrt(3)/5) = 19.106... degrees

For successive k, if adding this pushes the b^k angle past +30deg then the preceding digit goes past -30deg and becomes the new maximum X. Write the angle as a fraction of 60deg (pi/3),

    F = atan(sqrt(3)/5) / (pi/3)  = 0.318443 ...

This is irrational since b^k is never on the X or Y axes. That can be seen since 2/sqrt3*imag(b^k) mod 7 goes in a repeating pattern 1,5,4,6,2,3. Similarly 2*real(b^k) mod 7 so not on the Y axis, and also anything on the Y axis would have 3*k fall on the X axis. 🤯

Digits low to high are successive steps back cyclically 6,5,4,3,2,1 so that (with mod giving 0 to 5),

    N_xmax_digit(j) = (-floor(F*j+1/2) mod 6) + 1

The +1/2 is since initial direction b^0=1 is angle 0 which is half way between -30 and +30 deg.

Similarly for the location, using conj(w6) for rotation back

    z_xmax_exp(j) = floor(F*j+1/2)
                  = 0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5, ...
    z_xmax(k) = sum(j=0,k-1, conj(w6)^z_xmax_exp(j) * b^j)

By symmetry the maximum extent is the same in 60deg, 120deg, etc directions, suitably rotated. The N in those cases has the digits 1,2,3,4,5,6 cycled around for the rotation. In PlanePath triangular X,Y coordinates direction 60deg means when sum X+3*Y is a maximum, etc. 🔄

If the +1/2 in the floor is omitted then the effect is to find the maximum point in direction +30deg. In the PlanePath coordinates this means maximum sum S = X+Y.

    N_smax_digit(j) = (-floor(F*j) mod 6) + 1
                    = 1,1,1,1,6,6,6,5,5,5,4,4,4,3,3, ...

                 k-1
    N_smax(k) = digits N_smax_digit(j)    low digit j=0
                 j=0
              = 0, 1, 8, 57, 400, 14806, 115648, ...     decimal
              = 0, 1, 11, 111, 1111, 61111, 661111, ...  base7
    and also N_smax() + 1

    z_smax_exp(j) = floor(F*j)
                  = 0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6, ...
    z_smax(k) = sum(j=0,k-1, conj(w6)^z_smax_exp(j) * b^j)
              = 0, 1, 7/2+1/2*sqrt3*i, 9+3*sqrt3*i, 19+12*sqrt3*i, ...
    and also z_smax() + w6^2

    smax(k) = 2*real(z_smax(k)) + imag(z_smax(k))*2/sqrt3
            = 0, 2, 8, 24, 62, 172, 470, 1190, 3202, 8740, ...
              coordinate sum X+Y max

In the base figure, points 1 and 2 have the same X+Y=2 and this remains so in subsequent levels, so that for k>=1 N_smax(k) and N_smax(k)+1 are equal maximums. 💯

👀 SEE ALSO 👀

  • Math::PlanePath
  • Math::PlanePath::GosperIslands
  • Math::PlanePath::Flowsnake
  • Math::PlanePath::FlowsnakeCentres
  • Math::PlanePath::QuintetReplicate
  • Math::PlanePath::ComplexPlus

🏠 HOME PAGE 🏠

<http://user42.tuxfamily.org/math-planepath/index.html> 🌐

Okay so real talk, here's the boring-but-important part, and yes you do have to read it, it's giving ✨mandatory✨.

Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde

That's a whole decade of copyright years, we love longevity, we love consistency, periodt. 💅

This file is part of Math-PlanePath.

💖 The actual license vibes 💖

Math-PlanePath is free software — like, freedom free, not free-pizza free, iykyk. You can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version. Slay, options, we love choices. 🎀

⚠️ Bestie, listen. Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. So if it breaks, that's on the vibes, not on anybody legally. No cap. 💀

See the GNU General Public License for more details. Like, actually go look, it's right there.

You should have received a copy of the GNU General Public License along with Math-PlanePath. If not — and this is the part where you go 👀 — see <http://www.gnu.org/licenses/>.

🌈 Little reminder that if you didn't get a copy of the GPL with your copy of Math-PlanePath, that link up there is your bestie. Go get it. Fr fr.

🫶 And that's the whole page 🫶

That's it, that's the manpage, you made it to the end, iconic behavior, genuinely obsessed with you for reading all the way through. 🦋

Source/version: this page is for perl v5.32.0, dated 2021-01-23, and it's filed under the name Math::...licate(3pm) — the 3pm meaning it's a Perl module manpage, if you were wondering, and you're welcome. 👑

Go forth and plot those plane paths, main character. 💜✨🌸

📦 where this page comes from